This document summarizes a statistical model of a stock market consisting of buyers and sellers interacting on a lattice. Sellers are fixed in position with set prices, while buyers diffuse randomly and may purchase from sellers based on the difference between the seller's price and the buyer's valuation. The model could later be expanded to have sellers dynamically adjust prices based on aggregate demand, similarly to models of Brownian motion in financial markets. Statistical physics approaches are applied to understand this complex system of many interacting agents as an equilibrium process.

1. Simulation of a diffusion model describing
dynamic price information
Benjamin Yiwen Färber
May 7, 2014

2. Contents
1 Statistical stock market 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Specification of the model . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Relevance to the financial market . . . . . . . . . . . . . . . . . . 4
1.4 Agent’s behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 4
List of Figures 5
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3. Chapter 1
Statistical stock market
1.1 Introduction
Statistical physics describes phenomena consisting of very large numbers of par-
ticles or systems. One can think here of a classical or a quantum gas consisting
of very large numbers of molecules or atoms. To accurately describe this sys-
tem, it is isolated from the environment as interactions with the environment
leads to complex behaviour. Statistical physics does not need to be restricted
to the description of a gas but may be applied to dynamic phenomena like the
formation of gas bubbles in a boiling liquid. Lately it has been applied to other
non-physical contexts. The field of econophysics has emerged from address-
ing statistical physics methods and ideas to economic questions. The model of
Brownian motion has been used to study prices in stock markets. Stock mar-
ket bubbles and crashes were investigated in the framework of the Ising model
of magnetism. The main aspect in Econophysics are complex systems. Their
behaviour can be described as a result of a collective effect involving many in-
teracting particles. Forming an analogy of the stock market with molecular
systems, interactions of individual agents with their surroundings can be com-
pared to an ensemble of decisions in a statistical steady state, described by price
quotations in a stock market.
1.2 Specification of the model
A detailed definition of the model will be discussed here. The spatial distribu-
tion of particles is modeled on a quadratic lattice G = {0, 1, . . . , L − 1} of size
L × L with periodic boundary conditions. Two types of particles are denoted
as sellers and buyers. The sellers are fixed at their positions on the lattice and
are characterised, apart from their coordinates, by a certain price value p, at
which they sell. The buyers on the other hand diffuse over the lattice. They are
characterised by a value θ, which is their valuation. The difference between p
and θ defines the probability that a sale takes place. In the lattice gas model the
prices p and valuations θ are constant. Later, turning to a multi-agent system,
sellers change their prices p through a learning algorithm. The sellers might re-
spond to the aggregated individual demand similarly to the price settings found
in Brownian motion models for the stock market. This might be an equilibrium
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4. Figure 1.1: Sellers act as agents on a lattice, on which buyers, acting as agents,
move randomly, and occasionally buy at a seller when they hit the seller on a
lattice vertice.
process. As buyers are subject to the local neighbourhoods of individual sellers
the solution of the value function θ which is the expected equilibrium utility of
staying in the neighbourhood may be in direct analogy to the distribution of
the price variation δp.
1.3 Relevance to the financial market
The information affecting the dynamics of the price of a financial asset plays a
significant role in real financial markets. It may be of concern in this model as
sellers act upon information regarding their profit.
1.4 Agent’s behaviour
Learning is goal directed and can be modeled as a trial and error search. The
agent’s interaction with the environment is based on the valuation of the state
of the environment called the value function V (s) with s the state of the en-
vironment. The agent has the ability to plan the next action according to its
model of the environment. The value function may be implemented in form of
a temporal difference learning function,
V (s) ← V (s) + α (V (s ) − V (s)) (1.1)
where α is the step-size parameter. The value of a state s is updated to a
value regarding the difference of the values of the state s and s.
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5. List of Figures
1.1 Sellers act as agents on a lattice, on which buyers, acting as
agents, move randomly, and occasionally buy at a seller when
they hit the seller on a lattice vertice. . . . . . . . . . . . . . . . 4
5

6. Bibliography
[Sta04] Mantegna, R. N., Stanley, E., "An Introduction to Econophysics. Cor-
releations and Complexity in Finance."Cambridge University Press 2004
[Osb59] Osborne, M.F.M., "Browninan motion in the stock market" Operations
Research, Vol. 7 (1959), pp. 215-219
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